Conventionally, techniques have been proposed involving component analysis such as principal component analysis (including component analysis by sphereing), independent component analysis, and non-negative matrix factorization (NMF), wherein, given data strings, the data strings are separated into a plurality of components (such as principal components, independent components, or factors). For example, such technology is disclosed in a literature such as the following.    Patent Literature 1: Unexamined Japanese Patent Application KOKAI Publication No. 2003-141102
Patent Literature 1 proposes a technology in which principal component analysis or independent component analysis is used to separate a change in the quantity of a chemical substance into a plurality of components, and then uses these components to classify the factors of generation for the chemical substance into groups.
In general, independent component analysis involves the following:
(1) Upon receiving observed signals from m number of channels, the signals are conceived as an observed signal matrix X having m rows and T columns, wherein the time-axis observed values are arranged in rows in the column direction, and the rows are arranged in channel order.
The observed signal matrix X is then separated into:
(2) a source signal matrix S having n rows and T columns, wherein the time-axis source signal values of the source signal from n number of channels are arranged in rows in the column direction, and the n number of rows are arranged in channel order;
(3) a mixing matrix A having m rows and n columns and showing the path from the respective channels of the source signal to the respective channels of the observed signal; and
(4) a noise matrix N having m rows and T columns.
At this point, separation is performed such that for a predefined matrix operation c(•, •), the condition
(5) X=c(A, S)+N
is satisfied. Matrix multiplication, matrix convolutions, and various non-linear matrix link functions may be used as the matrix operation c(•, •).
In addition, while at this point it is typical to use iterative methods such as the gradient method, the conjugate gradient method, or Newton's method, the following is adopted as a cost function:
(6) a matrix function J(•, •, •) that takes a matrix of m rows and T columns, a matrix of m rows and n columns, and a matrix of n rows and T columns, and returns a scalar value.
More specifically, for J(X, A, S) wherein X is fixed and A and S are varied, a combination of A and S is calculated such that the value of J(X, A, S) is minimized (i.e., a minimum. Typically, the function solves for a maximum or a minimum, or in other words, an “extremum”.)
Examples of functions that can be used as this cost function include: a function J(X, A, S) that returns the absolute value of the element with the largest absolute value in the matrix (X−AS) (i.e., the maximum absolute value of the elements); a function that returns the average of the squares of each element in the matrix (X−AS); and a function that returns the total sum of the squares of each element in the matrix (X−AS). In addition, other constraints, such as non-negativity, sparseness, and statistical independence may be applied as necessary.